scholarly journals Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab

2015 ◽  
Vol 14 (4) ◽  
pp. 105-113 ◽  
Author(s):  
Stanisław Kukla ◽  
◽  
Urszula Siedlecka ◽  
Keyword(s):  
Heat Conduction ◽  
Laplace Transform ◽  
Problem Of Time ◽  
Fractional Heat Conduction

scholarly journals Heat conduction in a composite sphere - the effect of fractional derivative order on temperature distribution

2018 ◽  
Vol 157 ◽  
pp. 08008
Author(s):  
Urszula Siedlecka ◽  
Stanisław Kukla
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Laplace Transform ◽  
Time Derivative ◽  
Spherical Layer ◽  
Solid Sphere ◽  
Liouville Derivative ◽  
The Laplace Transform ◽  
Fractional Heat Conduction

The aim of the contribution is an analysis of time-fractional heat conduction in a sphere with an inner heat source. The object of the consideration is a solid sphere with a spherical layer. The heat conduction in the solid sphere and spherical layer is governed by fractional heat conduction equation with a Caputo time-derivative. Mathematical (classical) or physical formulations of the Robin boundary condition and the perfect contact of the solid sphere and spherical layer is assumed. The boundary condition and the heat flux continuity condition at the interface are expressed by the Riemann-Liouville derivative. An exact solution of the problem under mathematical conditions is determined. A solution of the problem under physical boundary and continuity conditions using the Laplace transform method has been obtained. The inverse of the Laplace transform by using the Talbot method are numerically determined. Numerical results show the effect of the order of the Caputo and the Riemann-Liouville derivatives on the temperature distribution in the sphere.

scholarly journals An analytical solution to the problem of time-fractional heat conduction in a composite sphere

2017 ◽  
Vol 65 (2) ◽  
pp. 179-186 ◽  
Author(s):  
S. Kukla ◽  
U. Siedlecka
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Thermal Contact ◽  
Time Derivative ◽  
Solid Sphere ◽  
Composite Sphere ◽  
Problem Of Time ◽  
Time Space ◽  
Fractional Heat Conduction

Abstract An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative.

Thermoelastic Interaction in an Infinite Long Hollow Cylinder with Fractional Heat Conduction Equation

2017 ◽  
Vol 9 (2) ◽  
pp. 378-392
Author(s):  
Ahmed. E. Abouelregal
Keyword(s):  
Heat Conduction ◽  
Laplace Transform ◽  
Hollow Cylinder ◽  
Heat Conduction Equation ◽  
Thermal Stresses ◽  
Generalized Thermoelasticity ◽  
Constant Heat Flux ◽  
Conduction Equation ◽  
The Laplace Transform ◽  
Fractional Heat Conduction

AbstractIn this work, we introduce a mathematical model for the theory of generalized thermoelasticity with fractional heat conduction equation. The presented model will be applied to an infinitely long hollow cylinder whose inner surface is traction free and subjected to a thermal and mechanical shocks, while the external surface is traction free and subjected to a constant heat flux. Some theories of thermoelasticity can extracted as limited cases from our model. Laplace transform methods are utilized to solve the problem and the inverse of the Laplace transform is done numerically using the Fourier expansion techniques. The results for the temperature, the thermal stresses and the displacement components are illustrated graphically for various values of fractional order parameter. Moreover, some particular cases of interest have also been discussed.

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